Schrödinger group

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The Schrödinger group is the symmetry group of the free particle Schrödinger equation.

Schrödinger algebra[edit]

The Schrödinger algebra is the Lie algebra of the Schrödinger group.

It contains Galilei algebra with central extension.

[J_i,J_j]=i \epsilon_{ijk} J_k,\,\!
[J_i,P_j]=i \epsilon_{ijk} P_k,\,\!
[J_i,K_j]=i \epsilon_{ijk} K_k,\,\!
[P_i,P_j]=0, [K_i,K_j]=0, [K_i,P_j]=i \delta_{ij} M,\,\!
[H,J_i]=0, [H,P_i]=0, [H,K_i]=i P_i.\,\!

Where J_i, P_i, K_i, H are generators of rotations (angular momentum operator), spatial translations (momentum operator), Galilean boosts and time translation (Hamiltonian) correspondingly. Central extension M has interpretation as non-relativistic mass and corresponds to the symmetry of Schrödinger equation under phase transformation (and to the conservation of probability).

There are two more generators which we will denote by D and C. They have the following commutation relations:

[H,C]=i D, [C,D]=-2i C, [H,D]=2i H,\,\!
[P_i,D]=i P_i, [K_i,D]=-iK_i,\,\!
[P_i,C]=-iK_i,[K_i,C]=0,\,\!
[J_i,C]=[J_i,D]=0.\,\!

The generators H, C and D form the sl(2,R) algebra.

The role of the Schrödinger group in mathematical physics[edit]

Though the Schrödinger group is defined as symmetry group of the free particle Schrödinger equation, it is realised in some interacting non-relativistic systems (for example cold atoms at criticality).

The Schrödinger group in d spatial dimensions can be embedded into relativistic conformal group in d+1 dimensions SO(2,d+2). This embedding is connected with the fact that one can get the Schrödinger equation from the massless Klein–Gordon equation through Kaluza–Klein compactification along null-like dimensions and Bargmann lift of Newton–Cartan theory.

References[edit]

  • C. R. Hagen, "Scale and Conformal Transformations in Galilean-Covariant Field Theory", Phys. Rev. D 5, 377–388 (1972)
  • Arjun Bagchi, Rajesh Gopakumar, "Galilean Conformal Algebras and AdS/CFT", JHEP 0907:037,2009
  • D.T.Son, "Toward an AdS/cold atoms correspondence: A geometric realization of the Schrödinger symmetry", Phys. Rev. D 78, 046003 (2008)

See also[edit]